Theorem imaeq2 5064

Description: Equality theorem for image. (Contributed by NM, 14-Aug-1994.)

Assertion

Ref	Expression
imaeq2	|- ( A = B -> ( C " A ) = ( C " B ) )

Proof of Theorem imaeq2

Step	Hyp	Ref	Expression
1		reseq2 5000	. . 3 |- ( A = B -> ( C |` A ) = ( C |` B ) )
2	1	rneqd 4953	. 2 |- ( A = B -> ran ( C |` A ) = ran ( C |` B ) )
3		df-ima 4732	. 2 |- ( C " A ) = ran ( C |` A )
4		df-ima 4732	. 2 |- ( C " B ) = ran ( C |` B )
5	2, 3, 4	3eqtr4g 2287	1 |- ( A = B -> ( C " A ) = ( C " B ) )

Syntax hints:   -> wi 4   = wceq 1395   ran crn 4720   |` cres 4721   "cima 4722

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084  df-opab 4146  df-xp 4725  df-cnv 4727  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732

This theorem is referenced by:  imaeq2i  5066  imaeq2d  5068  fimadmfo  5557  ssimaex  5695  ssimaexg  5696  isoselem  5944  f1opw2  6212  fopwdom  6997  ssenen  7012  fiintim  7093  fidcenumlemrk  7121  fidcenumlemr  7122  sbthlem2  7125  isbth  7134  ennnfonelemp1  12977  ennnfonelemnn0  12993  ctinfomlemom  12998  ctinfom  12999  tgcn  14882  iscnp4  14892  cnpnei  14893  cnima  14894  cnconst2  14907  cnrest2  14910  cnptoprest  14913  txcnp  14945  txcnmpt  14947  metcnp3  15185